The sentence is " There is the letter T in the end of the word and the letter C in the other end". Can you call the start of the word the other end?
The idiom is "one end" and "the other end."
You can omit "one end" only if you specified it earlier in conversation or context can fill in the details.
The "start" will always be whatever is the "one end".
This idiom is used for very long things, so you are implying this word is extremely long (it may be painted on a long billboard, for example).
There is the letter T in one end of the word and the letter C in the other end.